Heat a panini or sandwich press according to the manufacturer’s instructions. (Alternatively, heat a nonstick grill pan over medium-high heat.)
While the press is heating, combine the parsley, 1/4 cup of the olive oil, lemon juice, capers, garlic, anchovy, and a few grinds of pepper in a blender and blend until smooth.
Lay the cheese on the bottoms of the focaccia. Spoon the tuna over the cheese. Spread some of the salsa verde over the inside tops of the focaccia and put the tops on the sandwiches salsa verde side down. Brush both sides of the sandwiches with the remaining 2 tsp. oil.
Put the sandwiches on the press salsa verde side up, pull the top down, and cook until browned and crisp and the cheese is melted, 5 to 7 minutes, depending on how hot your machine is. (If using a grill pan, put a heavy pan on top of the sandwiches and cook, turning the sandwiches over once.) Carefully remove from the press and serve.